Glossary

Convergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is convergent if its sum or value approaches a finite, real number. This means the sum or area under the curve does not grow infinitely large.

Example:

The improper integral $\int_1^{\infty} \frac{1}{x^2} dx$ is convergent because it evaluates to a finite value of 1.

Decreasing Function

Criticality: 2

A function $f(x)$ is decreasing over an interval if its values are continuously getting smaller as $x$ increases. This is a necessary condition for applying the Integral Test.

Example:

The function $f(x) = \frac{1}{x^2}$ is a decreasing function on $[1, \infty)$ because its derivative $f'(x) = -\frac{2}{x^3}$ is always negative for $x \ge 1$ .

Divergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is divergent if its sum or value does not approach a finite number, often tending towards positive or negative infinity, or oscillating without limit.

Example:

The series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ is divergent because its corresponding improper integral $\int_1^{\infty} \frac{1}{\sqrt{x}} dx$ evaluates to infinity.

Improper Integral

Criticality: 3

An integral where at least one of the limits of integration is infinite, or the integrand has a discontinuity within the interval of integration. These are evaluated using limits.

Example:

The integral $\int_1^{\infty} \frac{1}{x} dx$ is an improper integral because its upper limit extends to infinity.

Indefinite Integral

Criticality: 2

The set of all antiderivatives of a function, represented by $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$ and C is the constant of integration.

Example:

The indefinite integral of $f(x) = \cos(x)$ is $\sin(x) + C$ , representing all functions whose derivative is $\cos(x)$ .

Infinite Series

Criticality: 3

The sum of the terms of an infinite sequence, typically written in summation notation, whose convergence or divergence is often determined by tests like the Integral Test.

Example:

The expression $\sum_{n=0}^{\infty} (\frac{1}{3})^n = 1 + \frac{1}{3} + \frac{1}{9} + \dots$ represents an infinite series.

Integral Test for Convergence

Criticality: 3

A test used to determine if an infinite series converges or diverges by comparing it to an improper integral of a related function. It applies when the function is positive, continuous, and decreasing.

Example:

To determine if the series $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges, one can use the Integral Test for Convergence by evaluating the improper integral $\int_1^{\infty} \frac{1}{x^2+1} dx$ .

Positive Function

Criticality: 2

A function $f(x)$ is positive over an interval if $f(x) > 0$ for all $x$ in that interval. This is a necessary condition for applying the Integral Test.

Example:

For the Integral Test, $f(x) = e^{-x}$ is a positive function on $[0, \infty)$ since all its values are greater than zero.

Sequence ($a_n$)

Criticality: 2

An ordered list of numbers, often defined by a formula, where each number corresponds to a positive integer index $n$. In the Integral Test, the terms of the series $a_n$ are derived from $f(n)$.

Example:

For the series $\sum_{n=1}^{\infty} \frac{n}{n^2+1}$ , the sequence is $a_n = \frac{n}{n^2+1}$ , generating terms like $\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \dots$ .

U-Substitution

Criticality: 3

A technique for evaluating integrals by simplifying the integrand through a change of variables, often used when the integrand contains a composite function and its derivative.

Example:

To evaluate $\int x e^{x^2} dx$ , one would use u-substitution by letting $u = x^2$ , which transforms the integral into a simpler form.

Glossary

Convergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is convergent if its sum or value approaches a finite, real number. This means the sum or area under the curve does not grow infinitely large.

Example:

The improper integral $\int_1^{\infty} \frac{1}{x^2} dx$ is convergent because it evaluates to a finite value of 1.

Decreasing Function

Criticality: 2

A function $f(x)$ is decreasing over an interval if its values are continuously getting smaller as $x$ increases. This is a necessary condition for applying the Integral Test.

Example:

The function $f(x) = \frac{1}{x^2}$ is a decreasing function on $[1, \infty)$ because its derivative $f'(x) = -\frac{2}{x^3}$ is always negative for $x \ge 1$ .

Divergent (Series/Integral)

Criticality: 3

An infinite series or improper integral is divergent if its sum or value does not approach a finite number, often tending towards positive or negative infinity, or oscillating without limit.

Example:

The series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ is divergent because its corresponding improper integral $\int_1^{\infty} \frac{1}{\sqrt{x}} dx$ evaluates to infinity.

Improper Integral

Criticality: 3

An integral where at least one of the limits of integration is infinite, or the integrand has a discontinuity within the interval of integration. These are evaluated using limits.

Example:

The integral $\int_1^{\infty} \frac{1}{x} dx$ is an improper integral because its upper limit extends to infinity.

Indefinite Integral

Criticality: 2

The set of all antiderivatives of a function, represented by $\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$ and C is the constant of integration.

Example:

The indefinite integral of $f(x) = \cos(x)$ is $\sin(x) + C$ , representing all functions whose derivative is $\cos(x)$ .

Infinite Series

Criticality: 3

The sum of the terms of an infinite sequence, typically written in summation notation, whose convergence or divergence is often determined by tests like the Integral Test.

Example:

The expression $\sum_{n=0}^{\infty} (\frac{1}{3})^n = 1 + \frac{1}{3} + \frac{1}{9} + \dots$ represents an infinite series.

Integral Test for Convergence

Criticality: 3

Example:

To determine if the series $\sum_{n=1}^{\infty} \frac{1}{n^2+1}$ converges, one can use the Integral Test for Convergence by evaluating the improper integral $\int_1^{\infty} \frac{1}{x^2+1} dx$ .

Positive Function

Criticality: 2

A function $f(x)$ is positive over an interval if $f(x) > 0$ for all $x$ in that interval. This is a necessary condition for applying the Integral Test.

Example:

For the Integral Test, $f(x) = e^{-x}$ is a positive function on $[0, \infty)$ since all its values are greater than zero.

Sequence ($a_n$)

Criticality: 2

An ordered list of numbers, often defined by a formula, where each number corresponds to a positive integer index $n$. In the Integral Test, the terms of the series $a_n$ are derived from $f(n)$.

Example:

For the series $\sum_{n=1}^{\infty} \frac{n}{n^2+1}$ , the sequence is $a_n = \frac{n}{n^2+1}$ , generating terms like $\frac{1}{2}, \frac{2}{5}, \frac{3}{10}, \dots$ .

U-Substitution

Criticality: 3

A technique for evaluating integrals by simplifying the integrand through a change of variables, often used when the integrand contains a composite function and its derivative.

Example:

To evaluate $\int x e^{x^2} dx$ , one would use u-substitution by letting $u = x^2$ , which transforms the integral into a simpler form.